Finite Group Databasebeta

← #27428488

Subgroups — #27428488

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Dic53C211yesmaximal, characteristic
2#20276278C211yesmaximal, characteristic, complemented
3#20276191C211yesmaximal, characteristic, complemented
4#20276151C211yesmaximal, characteristic, complemented
5#20276151C211yesmaximal, characteristic, complemented
6#20275290C211yesmaximal, characteristic, complemented
7#20275290C211yesmaximal, characteristic, complemented
8#20275008C211yesmaximal, characteristic
9#20274907C211yesmaximal, characteristic, complemented
10#20274907C211yesmaximal, characteristic, complemented
11#20274803C211yesmaximal, characteristic, complemented
12#20274801C211yesmaximal, characteristic, complemented
13#20274738C211yesmaximal, characteristic, complemented
14C2×#16810385C211yesmaximal, characteristic, complemented
15C53×#2486C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×D4×Dic53—21no—
18C22×C4⋊Dic53C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27#16810522—21no—
28#16810498—21no—
29#16810498—21no—
30#16810493C2211yescharacteristic
31#16810483C2211yescharacteristic
32#16810483C2211yescharacteristic
33C4×C4⋊Dic53C2211yescharacteristic
34#16810385C2211yescharacteristic
35#16810385C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#123×C53—21no—
49C2×C4×D4×C53C2211yescharacteristic
50C22×C4⋊C4×C53C2211yescharacteristic
51C53×#398—21no—
52C53×#397—21no—
53C53×#389C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#381—21no—
56C53×#380C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C53×#380C2211yescharacteristic
59C23×Dic53—21no—
60C23×Dic53—21no—
61C22×#88322—22no—
62C22×#88322—22no—
63C22×#88322—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68D4×Dic53—44no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53—41no—
76C2×C4⋊Dic53—41no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C4×Dic53—21no—
84#13133148—22no—
85#13133148—22no—
86#13133148—41no—
87#13133148—41no—
88C22⋊#88322—22no—
89C22⋊#88322—22no—
90C22⋊#88322—41no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94C22×D4×C53D411yescharacteristic
95C23×C4×C53—21no—
96C23×C4×C53C2311yescharacteristic
97C23×C4×C53C2311yescharacteristic
98#123×C53—44no—
99C4×D4×C53—22no—
100C4×D4×C53—22no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—22no—
105C2×C4⋊C4×C53—22no—
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C42×C53C2311yescharacteristic
117#95×C53—21no—
118#95×C53—21no—
119#95×C53—21no—
120#95×C53—21no—
121#95×C53—21no—
122C24×C53C2×D411yescharacteristic
123C24×C53C2×D411yescharacteristic
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53SD1614yes—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—42no—
142C22×C4×C53—42no—
143C22×C4×C53—42no—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53#4112yes—
149C22×C4×C53#4112yes—
150C22×C4×C53C2411yescharacteristic
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—42no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53—42no—
159C22×Dic53—81no—
160C22×Dic53—81no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53—42no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—42no—
176C22⋊C4×C53—42no—
177C22⋊C4×C53—42no—
178C42×C53—21no—
179C42×C53—21no—
180C22⋊Dic53—42no—
181C22⋊Dic53—42no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—42no—
190C4⋊Dic53—42no—
191C4⋊Dic53—42no—
192C4⋊Dic53—81no—
193C4⋊Dic53#4112yes—
194C4×Dic53—42no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—44no—
207C2×#88322#4112yes—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53#12312yes—
219C23×C53C2×SD1612yes—
220C23×C53C2×SD1612yes—
221C23×C53AGL(1,ℤ/8)12yes—
222C23×C53C22×D411yescharacteristic
223D4×C53—28no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—84no—
255C2×C4×C53#12312yes—
256C2×C4×C53C2×SD1612yes—
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53H(5,𝔽2)11yescharacteristic
262C2×C4×C532−1+411yescharacteristic
263C2×Dic53—22no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—84no—
276C2×Dic53—84no—
277#88322—22no—
278#88322—41no—
279#88322—41no—
280#88322—41no—
281#24937—531nomaximal, Sylow
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53#48114yes—
296C22×C53C2×#12311yescharacteristic
297C22×C53#54611yescharacteristic
298C22×C53#54711yescharacteristic
299C22×C53C22×SD1611yescharacteristic
300C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C53#57711yescharacteristic
302C22×C53#57811yescharacteristic
303C4×C53—22no—
304C4×C53—22no—
305C4×C53—41no—
306C4×C53—41no—
307C4×C53—81no—
308C4×C53—81no—
309C4×C53—81no—
310C4×C53—81no—
311C4×C53#52911yescharacteristic
312C4×C53#53911yescharacteristic
313C4×C53#55111yescharacteristic
314C4×C53#55211yescharacteristic
315Dic53—42no—
316Dic53—81no—
317Dic53—81no—
318Dic53—81no—
319#2486D5311yescharacteristic
320D4×C4⋊C4—531no—
321#2100—531no—
322#2084—531no—
323#2077—531no—
324#2077—531no—
325#1919—531no—
326#1919—531no—
327#1869—531no—
328#1853—531no—
329#1853—531no—
330#1824—531no—
331#1823—531no—
332#1807—531no—
333C2×#340—531no—
334C2×C53—24no—
335C2×C53—42no—
336C2×C53#241911yescharacteristic
337C2×C53C2×#48111yescharacteristic
338C2×C53#309711yescharacteristic
339C2×C53#329811yescharacteristic
340C2×C53#331011yescharacteristic
341C2×C53#333511yescharacteristic
342C2×C53#334311yescharacteristic
343C23×C8—531no—
344C2×#123—21no—
345C2×C4×D4—1061no—
346C2×C4×D4C2×D5311yescharacteristic
347C22×C4⋊C4—531no—
348C22×C4⋊C4—531no—
349C22×C4⋊C4C2×D5311yescharacteristic
350C2×#106—531no—
351C2×#106—531no—
352C2×#106—531no—
353C2×#106—531no—
354C2×C22⋊C8—531no—
355C2×C22⋊C8—531no—
356C2×C22⋊C8—531no—
357#398—21no—
358#397—21no—
359#389—1061no—
360#389C2×D5311yescharacteristic
361#389C2×D5311yescharacteristic
362#385—1061no—
363#385—1061no—
364#384—531no—
365#381—21no—
366#380—531no—
367#380—531no—
368#380C2×D5311yescharacteristic
369#380C2×D5311yescharacteristic
370#380C2×D5311yescharacteristic
371C4×C4⋊C4—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#336—531no—
385#336—531no—
386C53#2493711yescharacteristic, complemented, Sylow
387C22×D4C22⋊D5311yescharacteristic
388C23×C4—21no—
389C23×C4—1061no—
390C23×C4—1061no—
391C23×C4C22×D5311yescharacteristic
392C23×C4C22×D5311yescharacteristic
393C22×C8—531no—
394C22×C8—531no—
395C22×C8—531no—
396C22×C8—531no—
397C22×C8—1062no—
398C22×C8—1062no—
399C22×C8—2121no—
400#123—44no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—2124no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—1062no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—2121no—
420C2×C4⋊C4—2121no—
421C2×C4⋊C4C22⋊D5311yescharacteristic
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C4⋊C4C22×D5311yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4C22⋊D5311yescharacteristic
432C2×C22⋊C4C22⋊D5311yescharacteristic
433C2×C42—1061no—
434C2×C42C22×D5311yescharacteristic
435#106—1062no—
436#106—1062no—
437#106—2121no—
438#106—2121no—
439C22⋊C8—1062no—
440C22⋊C8—1062no—
441C22⋊C8—2121no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450C24C2×C22⋊D5311yescharacteristic
451C24C2×C22⋊D5311yescharacteristic
452C2×D4—22no—
453C2×D4—22no—
454C2×D4#1171992314yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—4241no—
477C22×C4—4241no—
478C22×C4—2122no—
479C22×C4—2122no—
480C22×C4—2122no—
481C22×C4—2122no—
482C22×C4#1171994712yes—
483C22×C4#1171994712yes—
484C22×C4C2×C22⋊D5311yescharacteristic
485C22×C4C2×C22⋊D5311yescharacteristic
486C22×C4C2×C22⋊D5311yescharacteristic
487C22×C4C2×C22⋊D5311yescharacteristic
488C22×C4C23×D5311yescharacteristic
489C2×C8—532no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—2124no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—532no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—4241no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—2122no—
532C22⋊C4—2122no—
533C42—21no—
534C42—21no—
535C42—2122no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23#1313319812yes—
547C23C2×#1171992312yes—
548C23C2×#1171992312yes—
549C23#1313324212yes—
550C23C22×C22⋊D5311yescharacteristic
551D4—28no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—84no—
583C2×C4—1062no—
584C2×C4—4244no—
585C2×C4—4244no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4#1313319812yes—
598C2×C4C2×#1171992312yes—
599C2×C4#41×D5311yescharacteristic
600C2×C4#41×D5311yescharacteristic
601C2×C4C2×#1171994711yescharacteristic
602C2×C4C2×#1171994711yescharacteristic
603C2×C4#1313326311yescharacteristic
604C2×C4#1313326411yescharacteristic
605C8—1062no—
606C8—2121no—
607C8—2121no—
608C8—2121no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22#1681093914yes—
623C22C2×#1313319811yescharacteristic
624C22#1681126111yescharacteristic
625C22#1681127211yescharacteristic
626C22C22×#1171992311yescharacteristic
627C22C2×#1313324211yescharacteristic
628C22#1681143111yescharacteristic
629C22#1681143811yescharacteristic
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—81no—
637C4—81no—
638C4—4241no—
639C4—4241no—
640C4—4241no—
641C4—2122no—
642C4#1681115011yescharacteristic
643C4#1681121711yescharacteristic
644C4#1681129611yescharacteristic
645C4#1681131011yescharacteristic
646C2—24no—
647C2—42no—
648C2#2027818811yescharacteristic
649C2C2×#1681093911yescharacteristic
650C2#2028467811yescharacteristic
651C2#2028657411yescharacteristic
652C2#2028673011yescharacteristic
653C2#2028707211yescharacteristic
654C2#2028715511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2486C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#2486D53
hypocenterC53#24937
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2486

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20276278C2semidirect
3. #20276191C2semidirect
4. #20276151C2semidirect
5. #20276151C2semidirect
6. #20275290C2semidirect
7. #20275290C2semidirect
9. #20274907C2semidirect
10. #20274907C2semidirect
11. #20274803C2semidirect
12. #20274801C2semidirect
13. #20274738C2semidirect
14. C2×#16810385C2semidirect
386. C53#24937semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
D4×C4⋊Dic532
#202762782
#202761912
#202761512
#202752902
#202750082
#202749072
#202748032
#202748012
#202747382
C2×#168103852
C53×#24862
#2493753